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(pdftex.def) Requested size: 202.3529pt x 151.76526pt. [3 ] [4] [5] [6] [7] [8] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) Here is how much of TeX's memory you used: - 8302 strings out of 479618 - 124686 string characters out of 2865027 - 980964 words of memory out of 3000000 - 25370 multiletter control sequences out of 15000+200000 + 8334 strings out of 479618 + 125207 string characters out of 2865027 + 981750 words of memory out of 3000000 + 25398 multiletter control sequences out of 15000+200000 413212 words of font info for 65 fonts, out of 3000000 for 9000 1141 hyphenation exceptions out of 8191 68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s - -Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (8 pages, 234173 bytes). + +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (8 pages, 248293 bytes). PDF statistics: - 93 PDF objects out of 1000 (max. 8388607) + 101 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 93ace2d..cb9bb4e 100644 Binary files a/PS2/doc.pdf and b/PS2/doc.pdf differ diff --git a/PS2/doc.synctex.gz b/PS2/doc.synctex.gz index 8676a1a..5463743 100644 Binary files a/PS2/doc.synctex.gz and b/PS2/doc.synctex.gz differ diff --git a/PS2/doc.tex b/PS2/doc.tex index 53c08b8..db73568 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -14,8 +14,8 @@ \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{lipsum} - \usepackage{enumitem} +\usepackage{cancel} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -171,29 +171,30 @@ legend({'Numerical','Analytical'},'Location','northwest'); The analytical solution is \begin{equation} - \Phi(x) = A sin \left( \frac{n \pi}{L} x \right) + \phi(x) = A sin \left( \frac{n \pi}{L} x \right) \label{eq:analytical} \end{equation} In order to normalise this equation it must conform to the following: \begin{equation} - \int_{0}^{L} \left| \Phi(x) \right| ^2 dx = 1 + \int_{0}^{L} \left| \phi(x) \right| ^2 dx = 1. \label{eq:normalise} \end{equation} - Using the following identity: + We use the following identity: \begin{equation} - foo + \int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}. \label{eq:integral_ident} \end{equation} - Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate to the above identity. + Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical}) + to the above identity. Evaluating the integral gives us the following relationship \begin{equation*} - \frac{1}{A^2} = \frac{1}{2} L - \frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right) - \frac{1}{2} \cdot 0 + \frac{L}{4n \pi} sin(0) + \frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right)} - \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} sin(0)} \end{equation*} From this, we find: @@ -224,7 +225,15 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1c \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - \lipsum[1] + + From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that + $\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where + $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$. + + Fro + \item %1cii \lipsum[1] \item %1ciii